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Halls theorem. Returning to the matchmaking scenario of Section 7.3, suppose we have a

Chapter 7, Problem 7.30

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QUESTION:

Halls theorem. Returning to the matchmaking scenario of Section 7.3, suppose we have a bipartitegraph with boys on the left and an equal number of girls on the right. Halls theorem saysthat there is a perfect matching if and only if the following condition holds: any subset S of boysis connected to at least |S| girls.Prove this theorem. (Hint: The max-flow min-cut theorem should be helpful.)

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QUESTION:

Halls theorem. Returning to the matchmaking scenario of Section 7.3, suppose we have a bipartitegraph with boys on the left and an equal number of girls on the right. Halls theorem saysthat there is a perfect matching if and only if the following condition holds: any subset S of boysis connected to at least |S| girls.Prove this theorem. (Hint: The max-flow min-cut theorem should be helpful.)

ANSWER:

Step 1 of 3

For a graph display style G equals open parentheses V comma E close parentheses, a bipartite graph is obtained by partitioning the vertices of the graph into two sets display style P and display style Q in such a way that all edges have their one endpoint lies in display style P and other endpoint lies in display style Q. According to Hall’s theorem, for a bipartite graph display style open parentheses V subscript 1 comma V subscript 2 comma E close parentheses with display style open vertical bar V subscript 1 close vertical bar equals open vertical bar V subscript 2 close vertical bar has a perfect matching if and only if:

display style for all S subset of or equal to V subscript 1 comma open vertical bar S close vertical bar less or equal than open vertical bar N open parentheses S close parentheses close vertical bar

 

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